Koeman pleased with 'difficult' win
Senegal winger Sadio Mane's first goal for the south-coast club was enough to secure a fourth consecutive win at St Mary's for Koeman's men, as they moved above Manchester City into second spot.
Southampton dished out an 8-0 hammering of Sunderland last weekend, but they were made to work considerably harder by Stoke seven days later.
And manager Koeman was impressed by the way his team went about securing another three points.
He said: "It was difficult because we didn't kill the game. We created eight good chances and if you don't score the second one it's difficult.
"But football is strange because the first half was much better than last week. I think it was one of the best first halves of the season.
"In the second half we sometimes lost possession too easily, but they didn't create a lot.
"I think we deserved to win but it was a difficult one, more difficult than last week and maybe nicer than last week because the lads showed great character and ambition."
Goalscorer Mane, one of several new signings to have impressed at St Mary's so far this term, thought he had opened his Southampton account last weekend, only for the goal to be awarded to Sunderland defender Patrick van Aanholt.
But there were no doubts this time around and Koeman was delighted to see him get off the mark.
He added: "I heard the news yesterday and I explained it to Mane [the decision to take the goal off him].
"Maybe that was the reason he came to the bench today to say 'This one, yes? They're not going to take it away from me again?'
"But he had a great performance. He's fast and he's a good player, but it's his attitude that is important to the team."